Impulse Response of Causal Systems

[AcidRainLiTE]

I am reading "Linear System Theory and Design" by Chen and he says (in what follows g(t,tau) is the impulse response function):

If a system is causal, the output will not appear before an input is applied. Thus we have Causal <==> g(t,tau) = 0 for t < tau.​

However, this seems incorrect to me. For a given tau, g(t,tau) represents the output of the system in response to a delta spike centered at t= tau. So, for t < tau, the input stimulating g(t,tau) is 0. The author then concludes from this that the output for t < tau must also be zero. But this is false. Consider for instance a system consisting of a logical NOT gate where your the output is the logical NOT of your input. g(t,tau) would not be zero for t<tau for this system. What am I missing?

 

[AlephZero]

Usually, a NOT gate would mean you were talikng about digital signals, not anlog (continuously variable) ones.

Even if you extend your idea to an "analog NOT gate" where the "output = 1 - input" or sometning similar, that is still not a linear system.

For a linear system, if the input is always zero, the output must also be always zero.

 

[AcidRainLiTE]

For a linear system, if the input is always zero, the output must also be always zero.

Do you mean that, for a linear system, if the input and state variables are zero then the output must also be zero?

My textbook defines a linear system as a system which satisfies the following (x denotes state variable, u input, y output):

Given

$$\begin{array}{l l} x_1(t_0) \\ u_1(t), \quad t \geq t_0 \\ \end{array} \} \rightarrow y_1(t), \quad t \geq t_0$$

and

$$\begin{array}{l l} x_2(t_0) \\ u_2(t), \quad t \geq t_0 \\ \end{array} \} \rightarrow y_2(t), \quad t \geq t_0$$

we have that for any $\alpha_1, \alpha_2$

$$\begin{array}{l l} \alpha_1 x_1(t_0) + \alpha_2 x_2(t_0) \\ \alpha_1 u_1(t) + \alpha_2 u_2(t), \quad t \geq t_0 \\ \end{array} \} \rightarrow \alpha_1 y_1(t) + \alpha_2 y_2(t), \quad t \geq t_0$$

I can see how you can deduce from this condition that, if both $x(t_0) = 0$ and $u(t_0) = 0$ then $y(t_0) = 0$. But I cannot see how to do it without the condition on the state variables.

 

[AlephZero]

Do you mean that, for a linear system, if the input and state variables are zero then the output must also be zero?

Yes - sorry if the lack of precision confused you.

 

[alphysicist]

Your understanding is correct. The definition provided by the author is not entirely accurate. While it is true that a causal system will not produce an output before an input is applied, the impulse response function g(t,tau) does not necessarily have to be zero for t < tau. As you mentioned, it is possible for the output to be non-zero even for t < tau, as in the example of a logical NOT gate. This is because the impulse response function represents the output of the system in response to a delta spike at a specific time, not necessarily the output for all times.

A more accurate definition of a causal system would be that the output of the system at any given time t is only dependent on the input at or before that time, and not on any future inputs. In other words, the output cannot "see into the future" and is only affected by past or present inputs. This definition is consistent with the definition of causality in physics, where a cause must precede its effect.

It is important to note that while g(t,tau) may not be zero for t < tau, it must still be zero for t > tau. This is because the output cannot be affected by inputs that have not yet occurred. So, in the case of a logical NOT gate, g(t,tau) would be zero for t > tau, since the output at any time after tau would only depend on the input at or before tau.

In summary, the definition provided by the author may be oversimplified and does not fully capture the concept of causality in linear systems. It is important to understand that while the output of a causal system will not appear before an input is applied, the impulse response function may not necessarily be zero for t < tau.